Need help on the syntax of a basic proof

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In summary, the conversation is about a student who has two questions on an assignment that requires them to write two ε-δ proofs. They understand the logic behind the ε-δ definition of a limit but are unsure how to write a formal proof. They share one of the questions which asks for an ε-δ proof for lim (2x+5) = -1 as x approaches -3. The student has some basic algebra background and knows that δ = ε/2. They ask for guidance on writing the proof and receive some helpful tips from another person in the conversation. The conversation ends with the student feeling more confident in their approach and ready to turn in their assignment.
  • #1
jokerthief
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I have two questions on an assignment that require me to write two ε-δ Proofs. I understand the logic behind the ε-δ definition of a limit but I've never been asked to write a proof before and there aren't any examples in our book. I understand the semantics but not the syntax of what I need to do.

One of the questions is:

Give an ε-δ proof that: lim (2x+5) = -1
x-->(-3)

I know from the definition of a limit that if 0<|x+3|<δ, then |2x+6|<ε. After doing some basic algebra, I know that δ = ε/2.

So how do I write a proof with this information? I know that I could probably just put what I know into words and get credit (in fact, my prof told me to do exactly that) but I want to know how to write a tight formal proof.
 
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  • #2
jokerthief said:
I have two questions on an assignment that require me to write two ε-δ Proofs. I understand the logic behind the ε-δ definition of a limit but I've never been asked to write a proof before and there aren't any examples in our book. I understand the semantics but not the syntax of what I need to do.

One of the questions is:

Give an ε-δ proof that: lim (2x+5) = -1
x-->(-3)

I know from the definition of a limit that if 0<|x+3|<δ, then |2x+6|<ε. After doing some basic algebra, I know that δ = ε/2.

So how do I write a proof with this information? I know that I could probably just put what I know into words and get credit (in fact, my prof told me to do exactly that) but I want to know how to write a tight formal proof.

Exactly what basic algebra did you do? I hope you didn't throw away your scratch paper! That's crucial to the proof!

I imagine what you did was start with |(2x+5)-(-1)|= |2x+ 6|= 2|x+ 3|< [itex]\epsilon[/itex] so |x+6|= |x-(-3)|< [itex]\epsilon[/itex]/2.

Now reverse that:
Take [itex]\delta= \epsilon/2[/itex]. If |x-(-3)|= |x+ 3|< [itex]\delta[/itex]= [itex]\epsilon[/itex]/2, then 2|x+3|= |2x+ 6|= |2x+5-(-1)|< [itex]\epsilon[/itex].

Actually, what you did initially is perfectly good and is what you will see in most textbooks. It is sometime called "synthetic" proof. Start from what you want to prove and go to something that is part of the hypothesis). As long as every step is reversible, the actual proof, going form the hypothesis to what you want to prove, is obvious and doesn't have to be stated.
 
  • #3
HallsofIvy said:
Exactly what basic algebra did you do? I hope you didn't throw away your scratch paper! That's crucial to the proof!

Ha ha, no, I'm not that much of a noob. I pretty much knew what to do but wasn't sure if I was correct. Your post helped confirm to me that I was going in the right direction. I was trying to make it more difficult than it actually was. Thanks HallsofIvy, your post helped me feel confident turning in my assignment today.
 

Related to Need help on the syntax of a basic proof

1. What is a proof in mathematics?

A proof is a logical and systematic way of showing that a statement or theorem is true. It involves starting with known facts and using logical reasoning to arrive at a conclusion.

2. What is the syntax of a basic proof?

The syntax of a basic proof typically involves stating the theorem or statement to be proven, listing any given information, and then providing a logical sequence of steps to arrive at the conclusion. This may include using definitions, axioms, and previously proven theorems.

3. How do I structure a proof?

A proof should be well-organized and clearly structured. It should include an introduction stating the theorem to be proven, a list of given information, a body of logical steps, and a conclusion that restates the theorem. It is also important to use proper mathematical notation and to clearly label each step.

4. What are common mistakes to avoid in a proof?

Some common mistakes to avoid in a proof include assuming what needs to be proven, using incorrect notation or definitions, and skipping steps without proper justification. It is also important to check for logical errors and to ensure that each step follows logically from the previous one.

5. How can I improve my skills in writing proofs?

Practice is key in improving your skills in writing proofs. It is also helpful to study examples of well-written proofs and to seek feedback from peers or a mentor. Additionally, understanding and reviewing basic mathematical concepts and logic can also aid in writing clear and concise proofs.

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